Modern Mathematics Form 4

Friday 12 June 2015

Mathematics chapter 7 - probability

1 - Definition

- Probability is the measure of how likely something will occur.
- It is the ratio of desired outcomes to total outcomes.
- (# desired) / (# total)
- Probabilities of all outcomes sums to 1.

2 - Example

- If I roll a number cube, there are six total possibilities. (1,2,3,4,5,6)
- Each possibility only has one outcome, so each has a PROBABILITY of 1/6.
- For instance, the probability I roll a 2 is 1/6, since there is only a single 2 on the number cube.

3 - Practice

- If I flip a coin, what is the probability I get heads?
- What is the probability I get tails?
- Remember, to think of how many possibilities there are.

4 - Answer

- P(heads) = 1/2
- P(tails) = 1/2
- If you add these two up, you will get 1, which means the answers are probably right.

5 - Two or more events

- If there are two or more events, you need to consider if it is happening at the same time or one after the other.

6 - “And”

- If the two events are happening at the same time, you need to multiply the two probabilities together.
- Usually, the questions use the word “and” when describing the outcomes.

7 - “Or”

- If the two events are happening one after the other, you need to add the two probabilities.
- Usually, the questions use the word “or” when describing the outcomes.

8 - Practice

- If I roll a number cube and flip a coin:
- What is the probability I will get a heads and a 6?
- What is the probability I will get a tails or a 3?

9 - Answers

- P(heads and 6) = 1/2 x 1/6 =1/12
- P(tails or a 5) = 1/2 + 1/6 = 8/12 = 2/3

10- Experimental Probability

- An experimental probability is one that happens as the result of an experiment.
- (# of outcomes) / (# of trials)
- The probabilities we have done so far are “theoretical probabilities”, because there was no experiment.

11- Experiment

- Flip a coin 50 times, and write down what happens for each flip.
- In the end, find the experimental probabilities by writing the how many times heads and tails occurred over the
- total number of trials (flips)

-Fatehah & Balqis-

Monday 1 June 2015

CHAPTER 11 : LINES AND PLANES IN 3-DIMENSIONS

CHAPTER 11

LINES AND PLANES IN 3-DIMENSIONS

11.1 : Angles between Lines and Planes
* Identifying planes

A plane is flat surface of an object.

A two-dimensional shape has two dimensions which are length and breadth,and has only one plane.This shape has only area does not have volume.

A three-dimensional shape has three dimensions which are length,breadth and height.It has more than one surface(planes or curved surfaces).This shape has both area and volume.

*Identifying horizontal,vertical and inclined planes

There are three types of planes :
(a) Horizontal plane - A plane that is parallel to the horizontal surface.

(b) Vertical plane - A plane that is perpendicular to the horizontal surface.

*Sketching three dimensional shapes

Three dimensional shapes can be drawn on grid papers or blank papers.The specific planes can then be identified as horizontal planes,vertical planes or inclined plane.

*Identifying lines that lie on or intersect with a plane

In the diagram below,the line AB lies on the plane EFGH. Every point on the line AB lies to plane .

In diagram below, the line CD intersects the plane KLMN. The line CD meets the plane at only one point.

* Identifying normals to a plane
A normal to a plane is a straight line which perpendicular to any line on the plane passing through the point of intersection of the line and the plane.

PQ is the normal to plane ABCD as shown below.

*Orthogonal projections

The orthogonal projection of a line PR on a plane,with point R on the plane,is the line joining R to the point of intersection of the normal from P to the plane,that is line RQ.

* The angle between a line and a plane

The angle between a line and a plane is the angle between the line and its orthogonal projection of the line on the plane.

* Solving problems

The solve problems involving the angles between a line and a plane follow

the steps below :

Identify the normal to the given plane and the orthogonal projection of given line on the plane.

Sketch the right-angled triangle involved.

Identify the angle between the line and the plane.

Solve the problem using Pythagoras' theorem and / or trigonometric ratios.

11.2 : Angle Between Two Planes

*The line of intersection between two planes

Two planes, PQRS and RSUT meet at a straight line,RS, which is known as the line of intersection between the two planes.

*Drawing perpendicular lines to the line of intersection of two planes

To draw perpendicular lines to the line of intersection of two planes follow

the steps below.

Draw the line of intersection of two planes.

Mark a point on the line of intersection

From the point, draw two lines, one on each plane which is perpendicular to the planes.

RS is the line of intersection between the two planes.

Line JK is on plane PQRS and perpendicular to RS.

Line KL is on plane RSTU and perpendicular to RS.

*The angle between two planes

The angle between two intersecting planes in the angle between two lines,one each plane,drawn respectively from one common point on the line of intersection and is perpendicular to the line of intersection.

In the diagram below,QR the line of intersection of the planes, PQR and QRST. PM and MN are perpendicular to the line QR at M.

Thursday 7 May 2015

Chapter 2 Quadratic Expressions and Equations

Generally, ${{x}^{2}}-6x+5=0$ is the quadratic equation, expressed in the general form of $a{{x}^{2}}+bx+c=0$ , where a=1, b=- 6 and c=5. The root is the value of x that can solve the equations.

A quadratic equation only has two roots.

Example1: What are the roots of ${{x}^{2}}-6x+5=0$ ?
Answer: The value of 1 and 5 are the roots of the quadratic equation, because you will get zero when substitute 1 or 5 in the equation. We will further discuss on how to solve the quadratic equation and find out the roots later.

1) Solve the quadratic equations

There are many ways we can use to solve quadratic equations such as using:

1)     substitution,
2)     inspection,
3)     trial and improvement method,
4)     factorization,
5)     completing the square and
6)     Quadratic formula.

However, we will only focus on the last three methods as there are the most commonly use methods to solve a quadratic equation in the SPM questions. Let’s move on!

Factorization

Factorization is the decomposition of a number into the product of the other numbers, example, 12 could be factored into 3 x 4, 2 x 6, and 1 x 12.

Example 2: Solve ${{x}^{2}}+7x+12=0$ using factorization.
Answer: We can factor the number 12 into 4 x 3. Remember, always think of the factors which can be added up to the get the middle value (3+4 = 7), refer factorization table below,

So we will get ( x + 3 )( x + 4 ) = 0,
x + 3 = 0 or x + 4 = 0
x = – 3 or x = – 4

Example 3: Solve $10x-3=8x^2$ using factorization.
Answer: Rearrange the equation in the form of $a{{x}^{2}}+bx+c=0$

So we will get (4x – 3)(2x – 1)=0,
4x – 3 = 0 or 2x – 1 = 0
x = $\frac{3}{4}$ or x = $\frac{1}{2}$

Completing the square

Example 4: Solve the following equation by using completing the square method.

Quadratic formula

Normally when do you need to use this formula?

1)      The exam question requested to do so!
2)     The quadratic equation cannot be factorized.
3)     The figure of a, b, and c of the equation $a{{x}^{2}}+bx+c=0$ are too large and hard to factorized.

Example 5: Solve $(x-2)=6x(x+3)$ using quadratic formula.

2) Form a quadratic equation

How do you form a quadratic equation if the roots of the equation are 1 and 2? Well, we can do the work out like this using the reverse method:

We can assume:

x = 1 or x = 2
x – 1 = 0 or x – 2 = 0

(x-1)(x-2)=0
x2-2x-x+2=0
x2-3x+2=0

So the quadratic equation is x2 – 3x + 2=0. This is the most basic technique to form up a quadratic equation.

Let’s assume we have the roots of $\alpha$ and $\beta$ :

In other words, we can form up the equation using the sum of roots (SOR) and product of roots (POR). If the roots are 1 and 2,

SOR = 1+2
= 3

POR = 1 x 2
=2

${{x}^{2}}-(\text{sum of roots)}x+(\text{product of roots)}=0$

${{x}^{2}}-3x+2=0$

Sometime we need to determine the SOR and POR from a given quadratic equation in order to find a new equation from a given new roots. In general form,

Let’s look at the example below on how the concept above can help us solve the question.

Example 6: Given that $\alpha$ and $\beta$ are the roots of $5{{x}^{2}}-2x-2=0$ , form a quadratic equation $a{{x}^{2}}+bx+c=0$ with the roots of ( $\alpha$ – 5 ) and ( $\beta$ – 5 ).

3) Determine the conditions for the type of roots

Refer back to example 2, we know that ${{x}^{2}}+7x+12=0$ has two different roots (-3 and -4) by solving using factorization method. However, how are we going to determine the types of roots of ${{x}^{2}}+7x+12=0$ without solving the equation? The trick is we can use ${{b}^{2}}-4ac$ .

${{b}^{2}}-4ac$ is called a discriminant. Remember, when the value is greater than 0, we have 2 different roots, when it is 0, we have 2 equal roots, and when it is less than 0, we have no roots.

From the quadratic equation, ${{x}^{2}}+7x+12=0$ , $a=1,b=7,c=12,\text{ so }{{7}^{2}}-4(1)(12)=1$ , we have 2 different roots since the discriminant is greater than zero. Refer table below.

Example 7: A quadratic equation ${{x}^{2}}+2hx+4=x$ has two equal roots. Find the possible values of h.

Done by ; swaggy anis and lame aniyah